A branch-and-bound algorithm for the minimum radius $k$-enclosing ball problem

TL;DR

This paper introduces a branch-and-bound algorithm to efficiently solve the NP-hard minimum $k$-enclosing ball problem by strategically exploring subset trees and employing a dual algorithm for subproblems.

Contribution

The paper proposes a novel branch-and-bound approach with an effective node ordering and a dual algorithm for subproblems to address the minimum $k$-enclosing ball problem.

Findings

The algorithm reduces the number of explored nodes significantly.

It efficiently finds the minimum radius ball containing at least $k$ points.

The approach is applicable in high-dimensional Euclidean spaces.

Abstract

The minimum $k$-enclosing ball problem seeks the ball with smallest radius that contains at least~$k$ of~$m$ given points in a general $n$-dimensional Euclidean space. This problem is NP-hard. We present a branch-and-bound algorithm on the tree of the subsets of~$k$ points to solve this problem. The nodes on the tree are ordered in a suitable way, which, complemented with a last-in-first-out search strategy, allows for only a small fraction of nodes to be explored. Additionally, an efficient dual algorithm to solve the subproblems at each node is employed.